LMFDB - Newform orbit 12.16.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [12,16,Mod(1,12)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("12.1"); S:= CuspForms(chi, 16); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	12.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$17.1232206120$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 2187 q^{3} + 45702 q^{5} + 1217888 q^{7} + 4782969 q^{9} - 26895924 q^{11} - 162581770 q^{13} - 99950274 q^{15} - 743272542 q^{17} - 4003014700 q^{19} - 2663521056 q^{21} - 30097540728 q^{23} - 28428905321 q^{25}+ \cdots - 128642370718356 q^{99}+O(q^{100}) $

q - 2187 * q^3 + 45702 * q^5 + 1217888 * q^7 + 4782969 * q^9 - 26895924 * q^11 - 162581770 * q^13 - 99950274 * q^15 - 743272542 * q^17 - 4003014700 * q^19 - 2663521056 * q^21 - 30097540728 * q^23 - 28428905321 * q^25 - 10460353203 * q^27 + 19021888926 * q^29 - 4621552936 * q^31 + 58821385788 * q^33 + 55659917376 * q^35 + 649297928654 * q^37 + 355566330990 * q^39 + 790230862890 * q^41 + 1388728387532 * q^43 + 218591249238 * q^45 - 3933841180608 * q^47 - 3264310329399 * q^49 + 1625537049354 * q^51 - 13472208095706 * q^53 - 1229197518648 * q^55 + 8754593148900 * q^57 - 24672598493364 * q^59 + 23630686395542 * q^61 + 5825120549472 * q^63 - 7430312052540 * q^65 + 32385083278292 * q^67 + 65823321572136 * q^69 - 74451150070920 * q^71 + 176524276453946 * q^73 + 62174015937027 * q^75 - 32756223088512 * q^77 - 137959485182488 * q^79 + 22876792454961 * q^81 - 458794939458348 * q^83 - 33969041714484 * q^85 - 41600871081162 * q^87 - 32239404369270 * q^89 - 198006386701760 * q^91 + 10107336271032 * q^93 - 182945777819400 * q^95 + 478308097627586 * q^97 - 128642370718356 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−2187.00

45702.0

1.21789e6

4.78297e6

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ +1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	12.16.a.a	✓	1
3.b	odd	2	1		36.16.a.a		1
4.b	odd	2	1		48.16.a.f		1
12.b	even	2	1		144.16.a.g		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
12.16.a.a	✓	1	1.a	even	1	1	trivial
36.16.a.a		1	3.b	odd	2	1
48.16.a.f		1	4.b	odd	2	1
144.16.a.g		1	12.b	even	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 45702 $ T5 - 45702 acting on $S_{16}^{\mathrm{new}}(\Gamma_0(12))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T + 2187 $ T + 2187
$5$	$ T - 45702 $ T - 45702
$7$	$ T - 1217888 $ T - 1217888
$11$	$ T + 26895924 $ T + 26895924
$13$	$ T + 162581770 $ T + 162581770
$17$	$ T + 743272542 $ T + 743272542
$19$	$ T + 4003014700 $ T + 4003014700
$23$	$ T + 30097540728 $ T + 30097540728
$29$	$ T - 19021888926 $ T - 19021888926
$31$	$ T + 4621552936 $ T + 4621552936
$37$	$ T - 649297928654 $ T - 649297928654
$41$	$ T - 790230862890 $ T - 790230862890
$43$	$ T - 1388728387532 $ T - 1388728387532
$47$	$ T + 3933841180608 $ T + 3933841180608
$53$	$ T + 13472208095706 $ T + 13472208095706
$59$	$ T + 24672598493364 $ T + 24672598493364
$61$	$ T - 23630686395542 $ T - 23630686395542
$67$	$ T - 32385083278292 $ T - 32385083278292
$71$	$ T + 74451150070920 $ T + 74451150070920
$73$	$ T - 176524276453946 $ T - 176524276453946
$79$	$ T + 137959485182488 $ T + 137959485182488
$83$	$ T + 458794939458348 $ T + 458794939458348
$89$	$ T + 32239404369270 $ T + 32239404369270
$97$	$ T - 478308097627586 $ T - 478308097627586
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	12.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
Significant digits:
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